There are a lot of different types of Japanese saws, most of the cut on the pull (which makes a lot more sense than pushing it) and they may have rigid backs or not, and they may or may not have set teeth, that is, the alternating bend in the teeth. The ones with no set can be used for cutting flush, and with care, even on finished wood, but they tend to jam in cuts of much depth. As with most things, there are really high-quality versions which can be extremely expensive, and really cheapies that would be sold at Home Depot, if they had such things in Japan.
"Back saw" as noted has nothing to do with cutting on the pull, it has to do with the stiffener, intended to make straight cuts for joinery. A smaller version is called a dovetail saw, imaginatively named because it is suitable for cutting dovetails, with finer teeth, and even stiffer. Some dovetail saws cut on the pull, and some people flip the blades around on "push" versions.
Brett